modeling of second harmonic generation in hole-doped ...modeling of second harmonic generation in...

Modeling of second harmonic generation in hole-doped silicon-germanium quantum wells for mid-infrared sensing

JACOPO FRIGERIO,1 ANDREA BALLABIO,1 MICHELE ORTOLANI,2,* AND MICHELE VIRGILIO3 1L-NESS, Dipartimento di Fisica, Politecnico di Milano, Polo Territoriale di Como, Via Anzani 42, I-

22100, Como, Italy 2Dipartimento di Fisica, Sapienza Università di Roma, Piazzale Aldo Moro 5, I-00185 Rome, Italy 3Dipartimento di Fisica”E. Fermi”, Università di Pisa, Largo Pontecorvo 3, I-56127 Pisa, Italy *[email protected]

Abstract: The development of Ge and SiGe chemical vapor deposition techniques on silicon

wafers has enabled the integration of multi-quantum well structures in silicon photonics chips

for nonlinear optics with potential applications to integrated nonlinear optics, however

research has focused up to now on undoped quantum wells and interband optical excitations.

In this work, we present model calculations for the giant nonlinear coefficients provided by

intersubband transitions in hole-doped Ge/SiGe and Si/SiGe multi-quantum wells. We employ

a valence band-structure model for Si1-xGex to calculate the confined hole states of

asymmetric-coupled quantum wells for second-harmonic generation in the mid-infrared. We

calculate the nonlinear emission spectra from the second-order susceptibility tensor, including

the particular vertical emission spectra of valence-band quantum wells. Two possible

nonlinear mid-infrared sensor architectures, one based on waveguides and another based on

metasurfaces, are described as perspective application.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Mid-infrared (MIR) photonics is receiving considerable attention due to the variety of

envisaged applications in medical diagnostics [1], biochemistry studies [2,3], chemical

analytics [4], and environmental monitoring for safety and security [5]. The interest towards

the MIR resides in the presence of unique vibrational fingerprints of molecules in the

wavelength range λ = 5 to 15 µm (photon energies from 80 to 240 meV). An important

feature of molecular sensing techniques in the MIR is the direct probing of quasi-Lorentzian

lineshapes of molecule absorption [6], as opposed to refractive index sensing, typically

performed at a single wavelength in the visible (VIS) or near-infrared (NIR) by detection of

angular shifts, as in surface plasmon resonance (SPR) sensors [7], or of phase shifts, as in

Mach-Zhender interferometers, either fiber-based [1] or integrated in photonic chips [5]. In

general, probing the molecular absorption lineshape (frequency, width, intensity and possible

asymmetry) leads to better molecular specificity and higher information quality than

refractive index sensing [3]. Measuring MIR spectral lineshapes is relatively straightforward

in the gas phase [8] or liquid phase [4], however it is still a challenge for thin solid films

deposited on a sensor surface [9]. Surface-enhanced transmission/reflection spectroscopy in

the MIR of ultrathin molecular films with Fourier Transform interferometers has developed

significantly in the last decade basing on plasmonic nanoantenna arrays [2,10,11],

metasurfaces [7,3] and evanescent wave sensors [12,13], but major issues on molecular

lineshape modification by near-field interactions and their conversion to detectable far-field

signals have not been solved yet [9].

An entirely different path could be followed by introducing nonlinear emission

spectroscopy in the MIR: with the radiation source positioned exactly at the molecule

Vol. 26, No. 24 | 26 Nov 2018 | OPTICS EXPRESS 31861

#340238 Journal © 2018

https://doi.org/10.1364/OE.26.031861 Received 20 Jul 2018; revised 10 Sep 2018; accepted 6 Oct 2018; published 19 Nov 2018

https://doi.org/10.1364/OA_License_v1https://crossmark.crossref.org/dialog/?doi=10.1364/OE.26.031861&domain=pdf&date_stamp=2018-11-19

location, the complex electromagnetic problem of multiple near-field/far-field conversions

[14] could be greatly simplified [15]. MIR tunable laser pump beams are now available for

use in future nonlinear photonic sensors in the MIR. Nonlinear frequency conversion in MIR

optical parametric oscillators is progressing steadily [16]. Quantum cascade lasers provide an

electrically pumped, continuous-wave platform for high-harmonic generation in the MIR [17–

19]. However, nonlinear optics in the MIR presents technological challenges related to optical

material transparency [20] and rescaling of VIS-NIR photonic architectures at longer

wavelengths, therefore the exploration of unconventional approaches to nonlinear optics is

much needed.

Besides classical approaches based on intrinsic nonlinearity of bulk crystals [21] and

novel approaches based e.g. on nonlinear plasmonics [22] and phononics [23], an approach

unique to the MIR is the exploitation of the giant nonlinearity of intersubband transitions

(ISBTs) in doped multi quantum wells (MQW). This phenomenon is based on the giant dipole

moment of a two-dimensional (2D) electron gas in a doped quantum well. In the simplest

condition of one single parabolic band and quantum confinement along the MQW crystal

growth direction z, the ground state is a 2D parabolic subband minimum occupied by the

electron gas with density N2D, and the excited states are 2D parabolic subbands at equal

energy difference for all electronic states. Therefore, N2D multiplies both the linear dipole

moment at a given inter-subband transition (ISBT) energy and the second-order nonlinear

tensor (2)

ijk (where the indexes i,j,k run across the two equivalent in-plane directions x, y and

the quantum confinement direction z). With typical values of N2D in the range of 1011 cm2,

the individual nonlinear coefficients, i.e. the elements of the tensor (2)

ijk , can reach values of

the order of 105 pm/V [17,18,24–27], three to four orders of magnitude higher than those

observed in the best nonlinear crystals. Similar behavior could be expected in principle for a

2D hole gas in the valence band, however the valence band structure is considerably more

complex than the conduction band, as the former features heavy hole (HH), light hole (LH)

and split off (SO) bands, while the latter features only one parabolic free electron band. The

intervalence-band mixing at nonzero wavevector k results in strong nonparabolic dispersion

of the hole states. In turn, the ISBT energy becomes a function of k, making the ISBT

spectrum more complex than that of a three-level system typically observed in electron-doped

MQWs. Nevertheless, a giant (2)

ijk can be designed for hole-doped MQWs by implementing

accurate models of the valence band beyond the simple parabolic approximation, as we do

here. Due to the peculiar selection rules of ISBTs involving both HH and LH states, the use of

valence-band quantum wells opens up the exciting perspective of surface emission of SHG

radiation by leveraging on the off-diagonal terms of the nonlinear susceptibility tensor )xzx

(2 [8,24,25].

The giant nonlinearity of ISBTs in MQWs was investigated more than two decades ago,

both theoretically [24] and experimentally [25,26]. It has been recently revisited using

metasurfaces [27] and plasmonic nanoresonators [17,18] providing strong field enhancement,

which is extremely useful to enhance nonlinear effects and in turn allows for ultrathin

nonlinear elements as opposed to relatively long interaction regions in integrated waveguides.

Recent studies of nonlinearity in doped MQWs have focused on electron-doped

InGaAs/AlInAs heterostructures (In1-xGaxAs wells and Al1-xInxAs barriers) on InP substrates

[17–19,27]. This material system represents an ideal platform for nonlinear MIR photonics

because it can be approximated with the ideal single-parabolic band model; it features a band

offset of 0.5 eV, quite higher than MIR photon energies of interest for molecular sensing; it

has a high refractive index of 3.6 useful for strong mode-confinement in waveguides; and it

can operate as active gain material to compensate for pump depletion and free-carrier losses

[19]. However, compound semiconductor heterostructures are not suited for the mass-

production of disposable sensing chips, which could instead be possible with silicon-foundry

compatible group-IV semiconductor heterostructures such as SiGe/Si (Si1-xGex wells and Si


barriers) and Ge/SiGe (Ge wells and Si1-xGex barriers). The indirect-bandgap disadvantage of

group-IV semiconductors does not play a role in unipolar MQW systems. Silicon photonic

chips now include MQW structures for quantum-confined Stark effect modulators [28],

quantum well IR photoionization (QWIP) detectors [29], integrated SiGe waveguides [30],

and all-semiconductor plasmonic nanoresonators [10]. At the same time, the nonlinearity of

ISBTs in doped Ge/SiGe and SiGe/Si MQWs is expected to be as large as that of their

compound semiconductor counterparts, but it has not been calculated nor measured so far. In

this work, we present an application study for MIR emission sensors based on second-

harmonic generation (SHG) in hole-doped MQWs made of Ge/SiGe and SiGe/Si whose band

offset of 0.5 eV enables MIR applications, unlike Ge/SiGe electron-doped MQWs that have

an offset around 0.1 eV [31]. The paper is organized as follows: in Section 2 the silicon-

germanium heterostructure material platform is introduced and three specific heterostructure

designs are described; in Section 3, a model for the giant nonlinearity of ISBTs in the valence

band of group-IV heterostructures is presented and SHG spectra are calculated; in Section 4,

the feasibility of epitaxial material growth is discussed; in Section 5, two nonlinear MIR

sensor architectures based on waveguide chips and plasmonic metasurfaces are proposed.

2. Material properties and model

MQWs in the valence band of Ge/Si systems can provide band offsets up to 0.5 eV for the

heavy hole band, comparable to those of InGaAs/AlInAs systems, a high refractive index up

to 4.0 for mode confinement and an extremely wide MIR transparency range especially in Ge-

rich MQWs. The band offsets obtainable for the light hole band are generally smaller,

depending on the composition and on the strain of the particular structure. The possibility for

loss compensation by electrical or optical pumping cannot be excluded in the future, as optical

gain has been predicted in Si/SiGe heterostructures [31]. The numerical modeling capability

and the knowledge of the material parameters for group-IV heterostructures [32,33] has

increased enormously in the last years. On these bases, we have constructed an accurate

model for the valence band of Si1-xGex heterostructures independently accounting for both

strain and composition at any x between 0 and 1. This model has been used to design three

heterostructures based on asymmetric coupled quantum wells (ACQWs) [14,17] suitable for

SHG because the asymmetry of the structure breaks the inversion symmetry of the Si1-xGex

crystal. In particular we propose a Ge-rich heterostructure (Ge/SiGe design) and a Si-rich

heterostructure (Si/SiGe design) that both exploit only HH to HH transitions for SHG; and

another Ge-rich heterostructure (GeLH) where HH to LH ISBTs are exploited instead for

SHG, enabling significant off-diagonal )xzx

(2 terms [24]. A scheme of the three designs is

shown in Fig. 1: the well with higher Ge content x is labeled as “main well (M)” and the other

well is labeled “secondary well (S)”. In all designs, the compressive strain of the well layer

guarantees that the HH state of the main well is the ground state of the system which we label

as HH0M. In this work, the series of excited HH states of the main well is labeled HH1M,

HH2M etc. while the series of states for the secondary well is labeled HH0S, HH1S etc., and

the LH and SO states of both wells are labeled accordingly. This nomenclature is justified by

the weak hybridization between states in the two wells, which therefore retain a strong

isolated-well character. The thickness of one period of the proposed ACQWs is between 12

and 16 nm, therefore, in order to fill the entire ridge height of a several μm thick MIR

waveguide up to 500 periods are required and strain simmetrization can be achieved inserting

undoped spacer layer of suitable concentration.


Fig. 1. Heterostructure layer scheme for the three different designs presented in this work. The

epitaxial layer sequence can be identically repeated a large number of times to increase the

thickness of the interaction region.

The different potential energy profiles at the valence band maximum (Γ point) for HH and

LH bands are shown in Fig. 2 for the three designs. In ACQWs, the potential energy profile

and the well thickness are both finely tuned in the two wells in order to engineer the energy

and wavefunction symmetry of the confined electron or hole states [24–26,31]. In Fig. 2, the

squared modulus of the wavefunctions of the confined states that participate to nonlinear

effects for the present designs are shown as thick black curves for HH states (Fig. 2(d) and

2(e)) and as thick red lines for LH states (Fig. 2(f)). All other states are shown as thin lines

with the same color code (green curves represent states in the split-off band). The ACQWs

have been designed to obtain three valence subband maxima at energies E0, E1, E2 with almost

equal spacing E01 = (E0 – E1) (E1 – E2) = E12, which is a requirement for SHG [17]. The

second harmonic emission takes place around E02 = (E0 – E2). The SHG related parameters for

the three ACQWs are reported in Table 1.

Table 1. Summary of nonlinear parameters of the ACQW structures for N2D =

5.0∙1011cm2.

Design E0E1E2 T )zzz

(2 )xzx(2

(meV)

peak (nm/V)

(meV)

peak (nm/V)

Ge/SiGe HH0MHH0SHH1M 10K 120 137 80 8.6

300K 119 38 69 12.2

Si/SiGe HH0MHH0SHH1M 10K 129 107 96 0.6

300K 129 35 67 1.5 GeLH HH0MLH0MLH0S 10K 78 38 94 8.9

300K 77 10 59 15.9


Fig. 2. (a,b,c) Valence band structure around the Γ point for the three heterostructure designs presented in this work. (d, e, f) Potential energy profile at the Γ point (straight lines) and

squared modulus of the wavefunctions (curves). Heavy hole states: black, Light hole states:

red, split-off band: green. The thicker curves indicate the states among which intersubband

transitions responsible for the giant nonlinearity take place.

3. Calculation of nonlinear emission spectra

We have performed the novel calculations of ISBT nonlinearities for the three designs

described in Fig. 1 and 2. As already pointed out for compound semiconductor materials, the

p-orbital like structure of the valence band provides high values of the non-diagonal term )

xzx

(2 of the nonlinear susceptibility tensor (2)ijk . At the same time, the value of the diagonal term (2)

zzz in the valence band should be of the same order (or higher) than the values calculated [33] and measured [34] for the electron-doped heterostructures materials that have

been experimentally investigated up to now. Note that the indexes i,j,k also correspond to the

required electric-field polarization directions of the output beam and of the two laser pump

beams, respectively. If j = k, then one single laser pump beam can be used for SHG. To

predict the second-order susceptibility related to ISBTs in the valence band, we rely on a

semi-empirical first-neighbor 3 5 *sp d s tight-binding Hamiltonian which includes spin-orbit

interaction (see [32,33] for details on the model) in order to calculate the electronic band

structure. In this framework, all the effects related to subband non-parabolicity and to the

momentum-dependence of the dipole matrix elements, which for p-type structures are

expected to play a relevant role, have been taken into account sampling the Brillouin zone

around the Γ point (see Fig. 1a-c). The Brillouin zone has been sampled also along the z

direction. The impact of the Hartree potential on the electronic wavefunctions is found to be

negligible for all doping levels considered, therefore calculations are performed in the flat-

potential approximation. In this way, we can estimate the non-linear susceptibility spectra as a

function of pump photon frequency ω to obtain the SHG efficiency at 2ω from (2)

ijk . This is

done by implementing Eq. (3).6.18) of Ref [35], where non-resonant contributions are also

considered. More precisely, assuming that only the fundamental subband at E0 is populated by

holes in the zero temperature limit (Fermi level crossing the HH valence band close to the Γ

point), we evaluate the third-order(2)

ijk as:


3(2) (0)

3 2

d(2 ) ( )

(2 )

( ) ( ) ( )

[( ( ) 2 ) )][( ( ) ) ]

( ) ( ) ( )

[( ( ) 2 ) )][( ( ) ) ]

( ) ( ) ( )

[( ( ) 2 ) )][(

ijk ll

lmn

i j k

ln nm ml

nl nl ml ml

i k j

ln nm ml

nl nl ml ml

k i j

ln nm ml

mn mn n

k

i i

i i

i

k

k k k

k k

k k k

k k

k k k

k ( ) ) ]

( ) ( ) ( )

[( ( ) 2 ) )][( ( ) ) ]

( ) ( ) ( )

[( ( ) 2 ) )][( ( ) ) ]

( ) ( ) ( )

[( ( ) 2 ) )][( ( ) ) ]

( ) (

l nl

j i k

ln nm ml

mn mn nl nl

j i k

ln nm ml

nm nm ml ml

k i j

ln nm ml

nm nm ml ml

k j

ln nm

i

i i

i i

i i

k

k k k

k k

k k k

k k

k k k

k k

k ) ( )

[( ( ) 2 ) )][( ( ) ) ]

( ) ( ) ( )

[( ( ) 2 ) )][( ( ) ) ]

i

ml

ml ml nl nl

j k i

ln nm ml

ml ml nl nl

i i

i i

k k

k k

k k k

k k (1)

In the above expression the index l runs over the two non-degenerate states of the

fundamental subband whose energy is ( )lE k while the m and n indices label the higher

energy valence states; (0) ( )ll k is the equilibrium hole distribution function of the level l and

( ) ( ) ( )lm l mE E k k k ; lm is related to the relaxation rate for the out-of-equilibrium density matrix element

nm and in our calculations we use a constant value, setting 5 meV, in agreement with the HWHM of ISBTs typically measured in SiGe QWs systems

[31,34]. Finally, dipole electric moments lm lm

e r are calculated within the semi-empirical

tight-binding approach following the procedure outlined in [32]. In general, a resonant peak is

obtained in (2)

ijk whenever one of the denominators of Eq. (1) almost vanishes, and the

corresponding numerator is nonvanishing. As it can be deduced from Eq. (1), a broadening of

the optical transition linewidth if compared to the discrete-level case (due to e.g. proximity of

the upper state to the top of the barrier, extreme nonparabolicity of the bands, …) may results

in the weakening of nonlinear response.In Fig. 3, the resulting )zzz

(2 spectrum is shown for the Ge/SiGe design at 10 K and at 300 K, for different values of N2D. The peak position is quite

stable against temperature. In Fig. 3a, the susceptibility spectrum features a dominant peak at

120 meV related to the HH0MHH0SHH1M transitions. The maximum calculated values

of the the nonlinear coefficient are 2∙104 pm/V for the chosen value γ = 5 meV, a value that

compares well with recent reports in InGaAs/AlGaAs systems [18] and is more than two

orders of magnitude higher than that of the best MIR nonlinear crystals. A weaker peak is

seen at 167 meV with significant nonlinear susceptibility value due to the resonance of the

HH0MLH0S transitions with the transitions from LH0S to LH1S, LH1M and SO0M states

(see Fig. 1(d)). The latter transitions involve LH states and therefore also give a nondiagonal

susceptibility component )xzx

(2 (not shown in Fig. 3), with significant peak value )xzx(2 = 8∙103

pm/V at T = 10 K, and )xzx

(2 = 3.3∙103 pm/V at T = 300 K and for N2D = 1.25∙1012 cm2. Notice that these numerical values strongly depend on the selected of the homogenous broadening γ


= 5 meV, and therefore they have to be verified by experiments, in which the actual

broadening may be different.

In Fig. 3, the increase of T distributes the holes into states at finite k around the Γ point,

where nonparabolicity plays a role hence making the spectral peak of (2)

ijk broader than it is at

low T. This reduces the peak value by a factor 3 for )zzz

(2 . In the insets, it is shown that the nonlinear coefficient increases proportionally to N2D up to 1∙1012 cm2, well within the

achievable range of present epitaxial material growth technology. Only at T = 10 K and for

N2D = 1.25∙1012 cm2 the nonlinear coefficients show a trend towards saturation, again due to

nonparabolicity effects arising for the filling of hole states at k 0.

Fig. 3. Diagonal term of the modulus of the susceptibility sensor(2)

zzz for the Ge/SiGe design at

10 K (a) and at 300 K (b), calculated for different free hole density levels N2D: blue, 2∙1011 cm2

(blue); 5∙1011 cm2 (cyan); 8∙1011 cm2 (green); 1.1∙1012 cm2 (gold); 1.3∙1012 cm2 (red). In the

inset, the peak value of the susceptibility(2)

zzz vs. N2D.

In Fig. 4, both the )zzz

(2 and )xzx(2 nonlinear coefficient spectra at 10 K and at 300 K are

reported for all three heterostructure designs. The main numerical values (peak energy and

nonlinear susceptibility value) are summarized in Table 1. According to [24], the dipole

orientation of the ISBTs involving both HH and LH states, associated with energy resonance

E01 E12, leads to peaks in )

xzx

(2 , while ISBTs involving only HH states give peaks in

the )zzz

(2 spectrum. The Ge/SiGe design features both a high )zzz(2 for laser pump frequency ω

120 meV due to the HH0MHH0SHH1M transitions and a high )xzx

(2 for ω 80 meV, due to the HH0MLH0MLH0S transitions, in addition to the peak at 167 meV, not shown

in Fig. 4 but shown in Fig. 3 instead. The Si/SiGe design features only one main peak in )zzz

(2 at ω 125 meV since the HH0MLH0MLH0S is not energy-resonant. The GeLH design

at T = 10 K shows, among other minor peaks, a dominant peak at ω 94 meV associated to

the energy resonance of the HH0MLH0MLH0S transitions. At room T, due to the

distribution of holes into states with k 0, the spectral features become more difficult to

interpret in terms of a simple three-level system because of the more complicated band

structure of the GeLH design depicted in Fig. 1(c). Nevertheless, the )xzx

(2 peak value value slightly increases with T although it is clearly redshifted far from the design value of 94 meV

obtained at T = 10 K. In general, designs with higher )zzz

(2 are expected to perform better in sensor configurations including metasurfaces and nanoantenna arrays, while designs with

higher )xzx

(2 will require two cross-polarized, counter-propagating pump beams in a waveguide, but they could lead to the relevant feature of surface emission [24], which could

be of high importance in novel MIR sensor architectures (see Section 5 below).


Fig. 4. Nonlinear susceptibility spectra for the three designs design at T = 10 K (a-c) and at T = 300 K (d-f) and for diagonal (black) and off-diagonal (red) tensor elements. The value of N2D =

5∙1011 cm2 has been considered.

4. Material growth issues

The epitaxial growth of Ge/SiGe and Si/SiGe heterostructures has undergone an impressive

development in the last decade driven by the envisioned applications in the silicon photonic

technology [28,30,36–39]. Nevertheless, the realization of Ge/SiGe and Si/SiGe quantum

wells with narrow wells, abrupt interfaces and high compositional mismatch between the well

and the barrier remains challenging. Unlike III-V semiconductors, Si and Ge are completely

miscible over the entire compositional range and, therefore, the realization of atomically sharp

interfaces is hindered by the natural tendency of the two elements to mix up. The intermixing

between Si and Ge is mediated by native point defects, namely vacancies and interstitials, as

in many other semiconductors, but the picture is further complicated by the influence of strain

and composition on the diffusion mechanisms. In order to estimate the robustness of the

proposed designs against the intermixing, we have repeated the calculation of the potential

profile and of the wavefunctions for the Ge/SiGe design in presence of intermixing, which has

been modelled for each well by a superposition of two error functions:

2-

1 2 ( )2 2 2

-b b w

ax

x x x x erf er

x

f

a

(2)

where xB and xW are the Ge fraction in the barrier and in the well respectively, a is the width

of the ideal box-like quantum well, and σ is the standard deviation of the intermixing density.


The length of the intermixed region (i.e. the distance over which the composition varies from

90% of xB to 90% of xW) can be conventionally defined as 3.3l . The quantum state

calculation results are reported in Fig. 5, together with those related to the ideal case. The

calculations have been performed by taking into account a realistic intermixing region length l

= 1.65 nm. The energy and the squared wavefunction of the HH states are shown in Fig. 5.

Clearly, the potential barriers do still exist and minor changes to the quantum states are visible

with respect to the ideal case. Experimental analysis will be required in the future to calculate

design corrections due to intermixing effects. Also, the effect of intermixing and alloy

scattering on coherence time of the electronic states is not considered here, and it may have a

major impact on line broadening hence on the intensity of nonlinear effects.

Fig. 5. Effect of the intermixing on the potential profile (ideal in black solid line, with

intermixing in red solid line) and on the wavefunctions (ideal in black dash-dot line, with

intermixing in red dashed line) for the Ge/SiGe design.

Fig. 6. Sketch of two possible sensing configurations. (a) integrated waveguides; (b) plasmonic

metasurfaces. The blue arrow indicates the incoming direction of the laser pump beam, while the red shaded areas represent the propagation volumes of the emitted SHG. D: detector, L:

lens, SL: substrate lens.


5. Perspectives and conclusions

There are two main practical configurations that have been employed to generate SHG in the

MIR: laterally-pumped waveguide chips [21,25,26] and, more recently, vertically-pumped

metasurfaces [17,18]. Both these configurations can be further engineered towards the

realization of a MIR sensor employing Ge-Si heterostructure chips to be fabricated in silicon

photonics foundries by lithography and dry etching. The silicon substrate chips could be

disposable or replaceable, and they could be inserted in an optical sensing system including a

pump laser in the long-wavelength infrared (LWIR, λ > 8 µm) and a MIR detector with an

integrated short wavelength-pass filter.

Traditionally, waveguides have been employed to observe nonlinear effects because they

guarantee long interaction regions, intrinsic spatial overlap between pump and output beams

and, in the case of single-mode waveguides, also a sizeable electric field enhancement factor

if compared to plane waves. The giant nonlinear response of MQWs and the strong absorption

coefficient of ISBTs imply that a few hundred microns of interaction length are usually

sufficient to deplete the pump beam and obtain maximum conversion efficiency [19].

Recently, state-of-the art MIR waveguides have been realized in both the Ge-on-Si and the

SiGe-on-Si platforms, exploiting the difference in refractive indexes between Ge (n = 4.0) and

Si (n = 3.4) in the MIR [39–42]. An example of a sensing configuration is shown in Fig. 6a,

where the pump beams propagate in the waveguide in a direction y parallel to the QW plane.

Both TM || zE and TE || xE polarizations are supported by the waveguide and the )xzx(2

produces a surface emitting SHG beam propagating along z with || xE . Phase matching issues

need to be thoroughly considered. The sensing volume is the space above the waveguide,

which is probed by the vertically-emitted SHG radiation before it travels to the detector, with

the target material absorbing radiation at frequency 2ω resonating with its vibrational mode in

the MIR.

Nonlinear metasurfaces consisting of two-dimensional arrays of plasmonic

nanoresonators, each of them containing the MQW layers in its sub-wavelength sized field-

enhancement regions, have also been considered [17,18,27]. Using vertical illumination, the

interaction length corresponds to the thickness of the MQW layers and therefore it is

extremely short. However, the strong field enhancement provided by the resonators can be

exploited to recover the high nonlinear efficiency, while also releasing selection rules and

phase-matching constraints. A silicon substrate lens (see Fig. 6b) is likely to be the most

suitable configuration to optimize SHG power collection. If the target material has a

vibrational fingerprint either at the fundamental frequency or at the second harmonic, it will

result in a measurable perturbation of the SHG process (typically, increase of losses and

decrease of SHG efficiency).

In conclusion, we have explored the giant nonlinearity effect in silicon-based hole-doped

Si1-xGex heterostructures whose growth process, if compared to more common III-V

semiconductor heterostructures, features cost-effective silicon foundry compatibility.

Exploiting a well established tight-binding approach, we were able to take into account

precisely the strong nonparabolicity of the different valence subband dispersion and the

orbital symmetry of the states involved in the intersubband transitions. The first design of

nonlinear frequency generation on silicon-based quantum wells are presented, and in

particular we have described three different designs of asymmetric coupled quantum wells for

second harmonic generation. Their practical realization and the robustness of the predicted

electronic structure against intermixing effects, unavoidable in Si1-xGex, has been critically

discussed. )zzz

(2 values in the mid-infrared range (pump photon energy between 50 meV and

170 meV) are found to be comparable to those predicted and measured in III-V quantum

wells, i.e. three orders of magnitude higher than the best nonlinear crystals. Furthermore,

owing to the interaction of the different valence bands (mostly heavy and light hole bands


with different orbital symmetry), significant values of the nondiagonal term of the

susceptibility tensor )xzx

(2 is predicted hence enabling surface-normal second-harmonic

emission, which is instead prevented in electron-doped quantum well systems. These results

pave the way for nonlinear light-generation devices based on silicon wafers emitting in the

mid-infrared range [43], to be used for applications such as sensors exploiting both

propagation in integrated germanium-rich waveguides and surface-normal emission

geometries.

Funding

Sapienza Università di Roma program “Bandi di Ateneo per la Ricerca 2017”.

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